Input-to-state stability in probability of constrained impulsive switched delayed systems with unstable subsystems

The problem of stochastic input -to -state stability in probability (ISSiP) is studied in this paper for a class of constrained impulsive switched systems with unstable subsystems, where the switching signal and impulsive signal are confined to the subsets of all modes. Especially, the impulsive intensity and impulsive density are both stochastic, which triggered by a renewal process. Taking into account asynchrony of switching and impulsive signal, a new comprehensive admissible edge -dependent average impulsive interval (CAED-AII) concept is proposed to characterize the impulsive signal. Under constrained switching and stochastic impulses, we investigate ISSiP of impulsive switched systems in two cases: the impulsive intensity is supposed to be stochastic and impulsive density is fixed, and the impulsive intensity and density are both stochastic. In these cases, the situation of the subsystem is divided into two scenarios for discussion as well. The first scenario is that all subsystems are unstable. The second scenario is that some subsystems are unstable and some subsystems are stable. With the help of the admissible edge -dependent average dwell time (AED-ADT), CAED-AII, and Lyapunov-like function, sufficient conditions are established for ISSiP of the whole system. Finally, two numerical examples are given to illustrate the developed results.